Sightseeing plan in London

Gud day to all

Im off to England from 3rd Aug to 10th Aug.
As Im gonna stay at my friend's place where is 2hours away from London for 3-4days, my time in London will be very limited :-(
Therefore, I need to pick up a couple of places in London to visit.
Im thinking I definitely need to visit Buckingham Palace and British museum.
Also,I might join one-day-tour to visit 3 places below: Stonehenge, windsor castle, Bath.
Any other places I must visit in London?

At the same time, Id like to know where there are some budget accommodation in London. Hopefully I wanna find the place which is convenient to come out London city. Are there any areas that is easy to come out to city and that have budget accommodation?

Thanx for your attention and have a lovely day.
Kazumi